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arXiv · 2005.02924

A short proof of the infinitesimal Hilbertianity of the weighted Euclidean space

Simone Di Marino·Danka Lučić·Enrico Pasqualetto
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Abstract

We provide a quick proof of the following known result: the Sobolev space associated with the Euclidean space, endowed with the Euclidean distance and an arbitrary Radon measure, is Hilbert. Our new approach relies upon the properties of the Alberti-Marchese decomposability bundle. As a consequence of our arguments, we also prove that if the Sobolev norm is closable on compactly-supported smooth functions, then the reference measure is absolutely continuous with respect to the Lebesgue measure.

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Simone Di Marino, Danka Lučić, Enrico Pasqualetto. 2020-05-06. A short proof of the infinitesimal Hilbertianity of the weighted Euclidean space. https://arxiv.org/abs/2005.02924

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